Consumption - Stanford University

1 Chapter 20 ConsumptionCharles I. Jones, Stanford GSBP reliminary, Comments WelcomeLearning Objectives:In this chapter, we study the neoclassical Consumption model, in which individualschoose the time path of theirconsumption to maximize utility. how this standard model leads to a benchmark solution in which Consumption is propor-tional to an individual s total wealth, including current financial wealth and the presentvalue of current and future labor income. the heterogeneity in consumer behavior at the micro level;some individuals, often therich, tend to follow the permanent income hypothesis, whileothers, often the poor, haveconsumption that is quite sensitive to current income. additional facts about Consumption in the aggregate, including the decline in the personalsaving rate and the rise in the debt-income ratio in recent Jones Consumption , November 25, 20092 Consumption is the sole end and purpose of all Adam Smith1.

2 IntroductionConsumption accounts for more than two thirds of GDP, more than $10 trillion dollars in economy. This spending results from the economic decisions of over 100 million house-holds as they purchase food, clothing, houses, vacations, refrigerators, cars, and health key economic forces shape their decisions?The models considered in this book until now treat Consumption in a very simple way. Inthe Solow model, individuals save a constant fraction of their income. In the main short-runmodel, people consume a constant fraction of potential this chapter, we develop what might be called theneoclassical Consumption model. In-dividuals choose Consumption at each point in time to maximize a lifetime utility function thatdepends on current and future Consumption . People recognize that income in the future maydiffer from income today, and such differences influence Consumption neoclassical model we explore in this chapter is a fundamental building block of mod-ern macroeconomics.

3 It is to Consumption what the Solow model is to the study of economicgrowth. This workhorse model allows us to develop a better, more intuitive understanding ofthe microfoundations of Consumption that were summarized earlier in Chapter 10. There, weoutlined the insights from the permanent income hypothesisof Milton Friedman and the lifecycle model of Consumption of Franco Modigliani. Here, we provide careful microfoundationsfor these frameworks and assess their empirical The Neoclassical Consumption ModelThe first insight of the neoclassical Consumption model is that one can make a great deal ofprogress by thinking of time as involving only two periods:todayand thefuture. People mayearn income today and in the future, they consume today and inthe future, and a key decisionthey have to make is how much to consume today versus in the future. This is the essence ofthe neoclassical Jones Consumption , November 25, 20093 The Consumption model then has two main elements: an intertemporal budget constraintand a utility function.

4 We discuss each of these in The Intertemporal Budget ConstraintConsider a consumer named Irving after Irving Fisher, one of the greatest economists of thefirst half of the twentieth century and one of the originatorsof the neoclassical consumptionmodel. Suppose that as of this moment, Irving has financial wealth equal toftoday. For example,this financial wealth would include Irving s saving accountbalance and his holdings of stocksand bonds. Irving earns labor incomeytodaytoday andyf uturein the future. Lettingcdenoteconsumption, Irving faces the following two budget constraints:ctoday=ytoday (ffuture ftoday)( )cfuture=yfuture+ (1 +R)ffuture.( )Both equations have the form Consumption equals income less saving. The first equationapplies to today, andffuture ftodayrepresents Irving s saving for the future the amount hesets aside to increase the balance in his financial second equation applies in thefuture, the second (and last) period of the model.

5 In this case, Irving earns labor incomeyfuturebut then also earns interest on his financial wealth. Becausethis is the last period of life, thereis nothing to save for and Irving consumes all of his income and wealth at that these two equations yields Irving sintertemporal budget constraint:1ctoday+cfuture1+R=ftoday+yto day+yfuture1+Rpresent value ofconsumption=financial wealth+human wealth total wealth( )This equation says that the present discounted value of Consumption must equal total is, Irving s Consumption is constrained by the total resources that will be available tohim in the present and in the future. These resources includehis existing financial wealthftoday. But they also include hishuman wealth the present discounted value of labor income,ytoday+yfuture1+ equation shows that Irving s Consumption in any given year can be very different from1 Rewrite the second equation asffuture= (yfuture cfuture)/(1 +R)and substitute this result into the first Jones Consumption , November 25, 20094his income.

6 Irving is allowed to save for the future if he so desires, but he can also borrow againsthis future labor income. What must be true is that the presentvalue of Consumption equals thepresent value of lifetime UtilityWe assume that Irving chooses his Consumption today and in the future in order to maximizeutility. For this to make sense, we have to explain how Consumption affects utility. The standardassumption in macroeconomics is that Consumption deliversutility through autility example, if Irving consumes some amountcin a given period, we assume he receivesu(c)units of utility, sometimescalled utils. We assume Irving gets more utility whenever consump-tion his higher, but that Consumption runs into diminishingreturns, often calleddiminishingmarginal utility. That is, each additional unit of Consumption raises utility by a smaller andsmaller amount. Diminishing marginal utility is quite intuitive and applies to all kinds of con-sumption.

7 The first night of the week eating dinner at a fancy restaurant is a special treat; afterten nights in a row, however, another night out seems much less desirable. An example of sucha utility function is shown in , and diminishing marginal utility is reflected in thecurvature of Irving consumes in two periods, utility needs to depend on Consumption today andon Consumption in the future. A natural way to express this iswith the following lifetime utilityfunction:U=u(ctoday) + u(cfuture).( )Irving s lifetime utility depends on how much he consumes today and on how much he con-sumes in the future. The parameter is some number such as or that captures theweight that Irving places on the future relative to today. For example, if = 1, then Irving treatsutility flows today and in the future equally. Alternatively, if <1, a given flow of utility is worthmore when it occurs Choosing Consumption to Maximize UtilityWe ve now completed the setup of the neoclassical Consumption model.

8 Irving gets utility fromconsuming in each period, as in equation ( ), and he must choose his Consumption to satisfythe intertemporal budget constraint in equation ( ). The model is closed by assuming Jones Consumption , November 25, 20095 Figure : Flow utilityu(c) Consumption , cUtility u(c)Note: A Consumption level ofcdelivers a flow of utility to the consumer ofu(c). Utilityrises whencincreases, but the amount of the increase gets smaller and smaller, reflectingdiminishing marginal choose his Consumption so as to maximize utility subject to his budget constraint:maxctoday,cfutureU=u(ctoday) + u(cfuture),subject toctoday+cfuture1 +R= W( )where we vedefined W ftoday+ytoday+yfuture1+R. That is, Wdenotestotal wealth, the sum of financialwealth and human this problem requires calculus, and the solution isderived step-by-step in the foot-note below. However, the solution turns out to be quite intuitive.

9 In fact, walking through theintuition will allow you to get the solution yourself without going through the , look at the utility function. If Irving consumes a little more today, the extra utility he2To solve the consumer s problem using calculus, begin by solving forcfutureusing the intertemporal budget con-straint:cfuture= (1+R)( W ctoday). Substituting this expression into the utility function, we can write the maximizationproblem in terms ofctodayonly:maxctodayu(ctoday) + u`(1 +R)( W ctoday) .We solve by setting the derivative of utility with respect toctodayequal to zero:u (ctoday) + u (cfuture)(1 +R)( 1) = this equation gives the solution in the main Jones Consumption , November 25, 20096gets is the marginal utility of Consumption today, which we can write asu (ctoday). Alternatively,Irving can consume a little more tomorrow, in which case he gets the marginal utility of con-sumption tomorrow, adjusted by the discount parameter: u (cfuture).

10 Now recall the logic of the intertemporal budget constraint. The essence of this constraintis that Irving can consume one unit today, or can save that unit and consume1 +Runits in thefuture. If he s maximized utility, Irving must be indifferent between consuming today or in thefuture. This key condition can be stated asu (ctoday) = (1 +R)u (cfuture).( )This expression is called theEuler equationfor Consumption . It is one of the most famousequations in macroeconomics, lying at the heart of advancedmacroeconomic models, and ithas a beautiful Euler equation essentially says that Irving must be indifferent between consuming onemore unit today on the one hand and saving that unit and consuming in the future on the Irving consumes today, he gets the marginal utility of Consumption today the left-handside of the equation,u (ctoday). If Irving saves that unit instead, he gets to consume1 +Runits inthe future, each giving himu (cfuture)extra units of utility.